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Coursenotes and Handouts
Week 1
- Lecture 1: Welcome to Math 19 (Coures Syllabus, Some Warmup Exercises)
- What is mathematics? What questions does it answer? What tools does it use?
- Lecture 2: An overview of functions (Coursenotes, Precal Review, Precal Worksheet)
- What is a function? What are some basic functions we will study? How can I put together two old functions to make a new function? What is function composition? Once I've put two functions together, how can I take them apart? What is the inverse of a function? What do I need to know about graphing functions? How can I tell if a given graph represents the graph of a function?
Week 2
- Lecture 3: Slope and Lines; Introduction to the tangent problem (Coursenotes, Handout)
- What are the properties of logs I'll need to know? How can I find the inverse of a composition of functions? How do I graph the inverse of f(x) if I'm given the graph of f(x)? What information do I need to write the equation of a line? What is calculus all about? What is the tangent problem? What problem do we need to solve in order to answer the tangent problem?
- Lecture 4: Limits of functions (Coursenotes)
- What is the tangent problem? What information do we need to solve the tangent problem? What does it mean to consider what happens to a function as inputs approach a fixed value? What is a limit? How can I find the limit of a function at a point given the graph of that function? How does the value of the function at a point affect the value of the limit of that function at that point? What are directional limits?
Week 3
- Lecture 5: Properties of limits (Coursenotes)
- How can I tell if a function has a limit based on directional limits? Are there ways for a function to not have a limit aside from directional limits disagreeing? What is the limit of a polynomial? If I know the limit of two functions, do I know the limit of their sum? difference? product? quotient? If two functions don't have a limit at a given point, can their sum have a limit at that point?
- Lecture 6: Continuity (Coursenotes, Handout)
- What does it mean to say a function is continuous at a? If a function is continuous at a, what can I say about the limit of the function at a or whether f is defined at a? How can I use continuity to make evaluating limits easier? If a function is undefined at a or doesn't have a limit at a, can I conclude that it isn't continuous at a? How do I evaluate limits of quotients when the denominator has nonzero limit? when the denominator has zero limit, but the numerator has nonzero limit? when the denominator and numerator both have limit 0?
- Lecture 7: The Intermediate Value Theorem (Coursenotes, Handout)
- What is `directional continuity' and how does it compare to directional limits? What types of discontinuity that I'm likely to encounter? I know what it means for a function to be continuous at a point, but what does it mean for a function to be continuous on an interval? What is the intermediate value theorem? How do I solve problems that starts `Show there exists a number c that ...' when I'm not even asked to say explicitly what that number is? Is there a number x so that cos(x) = x?
Week 4
- Lecture 8: The Derivative at a Point (Coursenotes)
- How can I solve the tangent problem? How do I compute the slope of the line tangent to a function f(x) at a point (a,f(a))? How do I write the equation of the line tangent to a function f(x) at a point (a,f(a))?
- Lecture 9: The Derivative at a point (Coursenotes)
- What is the derivative of a function at a point a? For a generic funtion f and a generic point a, what's the solution to the tangent problem?
Week 5
- Lecture 10: The derivative as a function (Coursenotes, Handout)
- What is the derivative of a function? How can I express the derivative as a limit? How can I understand the derivative geometrically? What is the derivative of a line? Given the graph of a function f(x), how do I sketch a graph of f'(x)? Given the graph of f'(x), how can I sketch a graph of f(x)?
- Lecture 11: The derivative as a function, Part II (Coursenotes)
- What are the typical ways in which a function can fail to be differentiable at a point? What is the derivative of sin(x), and how can we see this using the definition of the derivative as a limit? How painful is it to compute the derivative of the cube root of x using the definition of the derivative as a limit? What is the derivative of a sum of functions in terms of the derivatives of the functions themselves? What is the derivative of a scalar times a function in terms of the scalar and the derivative of the function? What is the power rule? What is the derivative of a function of the form xn?
- Lecture 12: Derivatives of Trigonometrics, Exponentials (Coursenotes)
- How can a function have a vertical tangent line? How can I spot the graph of f'(x) if I'm given the graph of f(x)? What is the derivative of ex? cos(x)? tan(x)? sec(x)? ax?
Week 6
- Lecture 13: The Product Rule and Quotient Rule (Coursenotes)
- How can I evaluate the derivative of a product of functions in terms of the derivatives of the factors? How can I evaluate the derivative of a quotient of functions in terms of the derivatives of the numerator and denominator?
- Lecture 14: The Chain Rule (Coursenotes)
- How can I evaluate the derivative of a composition of functions in terms of the derivatives of the constituent functions? How do I evaluate the derivative of a function which is the composition of 3 functions? 4 functions? Why is the chain rule called the chain rule?
- Lecture 15: Practice Computing Derivatives (Coursenotes, Handout)
- How can I use the chain rule to evaluate the derivative of ln(x)? arcsin(x)? More generally, how can I use the chain rule to evaluate the derivative of f-1(x) in terms of f(x) and its derivative f'(x)?
Week 7
- Lecture 16: Implicit Differentiation (Coursenotes, Handout)
- How can I solve the tangent problem for the circle? How can I solve the tangent problem for a graph which isn't the graph of a function? How can I use implicit differentiation to solve for the derivative of arcsin(x), arccos(x), and arctan(x)?
- Lecture 17: The Calculus/Geometry Dictionary (Coursenotes)
- Besides solving the tangent problem, what does a derivative do for me? What do the terms like increasing, decreasing, concavity, critical points, inflection points, and local extrema mean? What is the calculus/geometry dictionary? How do I connect information about the derivative of a function to information about the graph of the function?
Week 8
- Lecture 18: Local Extrema and Calculus (Coursenotes)
- How are local extrema of a function connected to calculus? How are critical points and local extrema related? Are all local extrema critical points? Are all critical points local extrema? What tests can I use to determine if a potential extreme value is an extreme value? What is the first derivative test? What are its pros and cons? What is the second derivative test? What are its pros and cons?
- Lecture 19: Practice with Application of Derivatives (Coursenotes)
- Given f(x) (but not its graph), how can I determine properties about its graph from information about its derivative?
Week 9
- Lecture 20: The Extreme Value Theorem (Coursenotes, Handout)
- What are absolute extrema of a function? When do I know if a function has absolute extremes? What are some examples of functions which do not have absolute extremes? If I have a continuous function on a closed interval, what procedure do I use to find absolute extremes? Can absolute extrema occur at places other than local extrema? How can I maximize a function of two variables?
- Lecture 21: Optimization Problems (Coursenotes, Handout)
- I have a word problem; how do I convert it to a calculus problem? How can I use the first derivative test to find absolute extrema? How do I optimize a continuous function on an open interval?
- Lecture 22: Practice with Optimization(Coursenotes)
- When do I know my problem is asking me to optimize a function on an open instead of a closed interval? What is the difference in technique between optimizing a continuous function on an open and a continuous function on a closed interval?
Week 10
- Lecture 23: Linearization (Coursenotes)
- What are the applications of derivatives we've seen in class so far? How can I use a line tangent to the graph of f(x) to approximate values of f(x)? How good are these approximations? Are they overestimates or underestimates, and what's a simple test for answering this question?
- Lecture 24: Course Overview (Coursenotes)
- What have we done this term? What has it bought for us?
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